Abstract: I will talk about a new kriging method that has a rational form. It is shown that the generalized least squares estimator of the mean from rational kriging is much more well behaved than that of ordinary kriging. Parameter estimation and uncertainty quantification for rational kriging are proposed using a Gaussian process framework. I will also discuss a generalized version of rational kriging, which includes ordinary and rational kriging as special cases. Extensive simulations carried out over a wide class of functions show that the generalized rational kriging performs on par or better than both ordinary and rational kriging in terms of prediction and uncertainty quantification. The potential applications of the new kriging methods in the emulation and calibration of computationally expensive models will be illustrated with real and simulated examples. The paper can be downloaded from https://doi.org/10.1080/01621459.2024.2356296 .